arXiv subjects

Théo Marty

Publications and source records attributed to Théo Marty.

9 recordsLinked to original sources

arXiv2026

Partial section III: for Anosov flows

In the previous papers in the series, we characterized partial cross-sections for general flows, in the spirit of Fried's work on global cross-sections. In this paper, we deduce several consequences for Anosov flows. We provide a homology criterion for the existence of a partial cross-section in a given cohomology class. Additionally, there are at most finitely many partial cross-sections in that cohomology class. We deduce that on a 3-dimensional hyperbolic manifold, any Anosov flow is homologically full.

math.DS↗

arXiv2025

Partial section II: classification for general flows

This is the second article in a series that aims at classifying partial sections of flows, that is a general family of transverse surfaces. In this part, we classify partial cross-sections for all continuous flows, in the spirit of Schwartzman-Fried-Sullivan theory. We give a dynamical criterion for the existence of partial cross-sections, which is a direct consequence of part I of the series. Then we describe all partial cross-sections using a cohomological criterion, resembling Fried's criterion. We also characterize the cardinality of the set of partial cross-sections in a given cohomology class.

math.DS↗

arXiv2025

Partial section I: $α$-recurrence and equivariant Lyapunov maps

This is the first article in a series that aims at classifying partial sections of flows, that is a general family of transverse surfaces. In this part, we deal with the dynamical aspect of the question. Given a flow on a compact manifold $M$ and a cohomology class $α$ of rank 1, we give a criterion for the existence of an $α$-equivariant Lyapunov map on an Abelian covering of $M$ associated to $α$. One important aspect of the existence of such Lyapunov maps, and of the classification of partial sections, is a type of recurrence set relative to $α$. We describe how that set depends on $α$.

math.DS↗

arXiv2025

From pre-lamination to foliated plane

To a singular foliation on the plane corresponds a circular boundary at infinity endowed with a pre-lamination on the circle. We solve the converse direction. We determine which pre-lamination on the circle are boundary at infinity of a foliation, and we build the corresponding (unique) foliation. We consider both regular foliations and a singular foliations with prong singularities.

math.DS↗

arXiv2025

Hitchin grafting representations I: Geometry

We give a geometric interpretation of Fock--Goncharov positivity and show that bending deformations of Fuchsian representations stabilize a uniform Finsler quasi-convex disk in the symmetric space $\mathrm{PSL}_d(\mathbb R)/\mathrm{PSO}(d)$.

math.DG↗

arXiv2025

Hitchin grafting representations II: Dynamics

The Hitchin component of the character variety of representations of a surface group $π_1(S)$ into $\mathrm{PSL}_d(\mathbb{R})$ for some $d \geq 3$ can be equipped with a pressure metric whose restriction to the Fuchsian locus equals the Weil--Petersson metric up to a constant factor. We show that if the genus of $S$ is at least $3$, then the Fuchsian locus contains quasi-convex subsets of infinite diameter for the Weil--Petersson metric whose diameter for the path metric of the pressure metric is finite. This is established through showing that biinfinite paths of bending deformations have controlled bounded length.

math.DG↗

arXiv2024

Skewed Anosov flows are orbit equivalent to Reeb-Anosov flows in dimension 3

We prove that in dimension 3, Anosov flows which are $\mathbb{R}$-covered and skewed are orbit equivalent to Reeb-Anosov flows. We characterize the existence of an invariant contact form or of a Birkhoff section with a given boundary, in terms of linking numbers between two invariant signed measures. Furthermore, we prove the existence of open book decompositions with one boundary component for Reeb-Anosov flows.

math.DS↗

arXiv2024

Oriented Birkhoff sections of Anosov flows

This paper gives 3 different proofs (independently obtained by the 3 authors) of the following fact: given an Anosov flow on an oriented 3 manifold, the existence of a positive Birkhoff section is equivalent to the fact that the flow is $\mathbb{R}$-covered positively twisted.

math.DS↗

arXiv2020

First-return maps of Birkhoff sections of the geodesic flow

This paper compares different pseudo-Anosov maps coming from different Birkhoff sections of a given flow. More precisely, given a hyperbolic surface and a collection of periodic geodesics on it, we study those Birkhoff sections for the geodesic flow on the unit bundle to the surface bounded by the collection. We show that there is a canonical identification of all those surfaces, and that the first-return maps induced by the flow can all be expressed as a composition of negative Dehn twists along the same family of curves: only the order depends on the choice of a particular surface.

math.GT↗